Dense dislocations enable high-performance PbSe thermoelectric at low-medium temperatures

PbSe-based thermoelectric materials exhibit promising ZT values at medium temperature, but its near-room-temperature thermoelectric properties are overlooked, thus restricting its average ZT (ZTave) value at low-medium temperatures. Here, a high ZTave of 0.90 at low temperature (300–573 K) is reported in n-type PbSe-based thermoelectric material (Pb1.02Se0.72Te0.20S0.08−0.3%Cu), resulting in a large ZTave of 0.96 at low-medium temperatures (300–773 K). This high thermoelectric performance stems from its ultralow lattice thermal conductivity caused by dense dislocations through heavy Te/S alloying and Cu interstitial doping. The dislocation density evaluated by modified Williamson-Hall method reaches up to 5.4 × 1016 m−2 in Pb1.02Se0.72Te0.20S0.08−0.3%Cu. Moreover, the microstructure observation further uncloses two kinds of dislocations, namely screw and edge dislocations, with several to hundreds of nanometers scale in length. These dislocations in lattice can strongly intensify phonon scattering to minimize the lattice thermal conductivity and simultaneously maintain high carrier transport. As a result, with the reduced lattice thermal conductivity and optimized power factor in Pb1.02Se0.72Te0.20S0.08−0.3%Cu, its near-room-temperature thermoelectric performance is largely enhanced and exceeds previous PbSe-based thermoelectric materials.

where the Fn(η) is the n-th order Fermi integral:   n n - where kB is the Boltzmann constant, e is the electron charge and Ef denotes the Fermi level, r is the scattering factor, and the acoustic phonon scattering has been assumed as the main carrier scattering mechanism with r=-1/2.
where Cp, ph and Cp, D denote heat capacity originated from phonon and lattice dilation, respectively. The phonon heat capacity Cp, ph can be calculated by: where ΘD is Debye temperature, R is molar gas constant, x=ℏω/kBT, in which ℏ and S3 denote reduced Planck constant and phonon vibration frequency, respectively.
The effects of lattice dilation on heat capacity Cp, D can be obtained from: where B is the isothermal bulk modulus, α is the linear coefficient of thermal expansion, and ρ is sample density. Notably, the electron heat capacity is not taken into account due to its negligible effects on lattice dilation compared with phonon.
Callaway model to predict lattice thermal conductivity. Callaway model shows the ratio of the conductivities of material containing defects to that of pure material 1-3 : in which κlat and κlat, p are the lattice thermal conductivities of the defected and parent materials, respectively, and κlat, p=1.68 W m -1 K -1 in this work. u is defined as 2, 4 : where Ω and h are the average atom volume and Planck constant, respectively. The Debye temperature (D), average sound velocity (va) and can be written as 5 : where kB, vl and vs are Boltzmann constant, longitudinal and shear sound velocities, respectively. In this work, there is no change on the sites of Pb after Q (Te or S) substituting Se sites, namely the imperfection scattering parameter ГPb=0, Thus, the where M is molar mass. Meanwhile, Г is a weighted sum of the mass fluctuation (ГM) and strain field fluctuation (ГS), can be written as 3,7 : in which ɛ is a phenomenological adjustable parameter related to the Poisson ratio (vp) and Grüneisen parameter (γ). Moreover, they can be expressed by 8,9 : where G is a ratio between the relative change of bulk modulus and banding length.
And ГM (Se, Q) and ГS (Se, Q) in equation (S14) can be expended as follows: where ΔM, Δr and r can be written as 7 : Then we can obtain: Modified Williamson-Hall method to calculate dislocation density. In consideration of that size and strain broadening are diffraction order independent and independent, respectively, Williamson and Hall suggested that the full width at half-maximum (FWHM) of line profiles can be written as [10][11][12] : is the strain contribution to line broadening and d is the average grain size or particle size.
,  and  are the diffraction angle and the wavelength of X-rays, respectively. When strain is caused by has the following form [13][14][15] : where A and A' are parameters determined by the effective outer cutoff radius of dislocations, Re, and the auxiliary parameters R1 and R2, respectively. ND * and Q * are the formal values of dislocation density and the correlation factors, respectively, they are related to the true values ND and Q as 13-15 :